One plus two-body random matrix ensembles with parity: Density of states and parity ratios

摘要

One plus two-body embedded Gaussian orthogonal ensemble of random matriceswith parity [EGOE(1+2)-$\pi$] generated by a random two-body interaction(modeled by GOE in two particle spaces) in the presence of a mean-field, forspinless identical fermion systems, is defined, generalizing the two-bodyensemble with parity analyzed by Papenbrock and Weidenm\"{u}ller [Phys. Rev. C{\bf 78}, 054305 (2008)], in terms of two mixing parameters and a gap betweenthe positive $(\pi=+)$ and negative $(\pi=-)$ parity single particle (sp)states. Numerical calculations are used to demonstrate, using realistic valuesof the mixing parameters appropriate for some nuclei, that the EGOE(1+2)-$\pi$ensemble generates Gaussian form (with corrections) for fixed parity eigenvaluedensities (i.e. state densities). The random matrix model also generates manyfeatures in parity ratios of state densities that are similar to thosepredicted by a method based on the Fermi-gas model for nuclei. We have alsoobtained, by applying the formulation due to Chang et al [Ann. Phys. (N.Y.){\bf 66}, 137 (1971)], a simple formula for the spectral variances defined overfixed-$(m_1,m_2)$ spaces, where $m_1$ is the number of fermions in the +veparity sp states and $m_2$ is the number of fermions in the -ve parity spstates. Similarly, using the binary correlation approximation, in the dilutelimit, we have derived expressions for the lowest two shape parameters. Thesmoothed densities generated by the sum of fixed-$(m_1,m_2)$ Gaussians withlowest two shape corrections describe the numerical results in many situations.The model also generates preponderance of +ve parity ground states for smallvalues of the mixing parameters and this is a feature seen in nuclear shellmodel results.